726 Mb. STOKES'S DISCUSSION OF A DIFFERENTIAL EQUATION 



For the sake of greater generality I shall suppose the travelling weight, instead of being free, to 

 be attached in this manner to a carriage. 



Let M be the mass of the weight, including the arm, k the radius of gyration of the whole 

 about the fulcrum, h the horizontal distance of the centre of gravity from the fulcrum, / the hori- 

 zontal distance of the point of contact of the weight with the bridge, w, y the coordinates of that 

 point at the time t, f, rj those of any element of the bridge, R the reaction of the bridge against 

 the weight, M' the mass of the bridge, R', R" the vertical pressures of the bridge at its two extremi- 

 ties, diminished by the statical pressures due to the weight of the bridge alone. Suppose, as 

 before, the deflection to be very small, and neglect its square. 



By D'Alembert's principle the effective moving forces reversed will be in statical equilibrium 

 with the impressed forces. Since the weight of the bridge is in equilibrium with the statical pres- 

 sures at the extremities, these forces may be left out in the equations of equilibrium, and the only 

 impressed forces we shall have to consider will be the weight of the travelling body and the reactions 



M' 

 due to the motion. The mass of any element of the bridge will be — rf^ very nearly ; the 



horizontal effective force of this element will be insensible, and the vertical effective force will be 



^d^, and this force, being reversed, must be supposed to act vertically upwards. 



The curvature of the bridge being proportional to the moment of the bending forces, let the 

 reciprocal of the radius of curvature be equal to K multiplied by that moment. Let A, B be the 

 extremities of the bridge, P the point of contact of the bridge with the moving weight, Q, any point 

 of the bridge between A and P. Then by considering the portion A Q of the bridge we get, taking 

 moments round Q, 



-$-'l'''f4'rS'<f-f>4 <*" 



»)' being the same function of ^' that ij is of ^. To determine K, let S be tlie central statical 

 deflection produced by the weight Mg resting partly on the bridge and partly on the fulcrum, which 



is equivalent to a weight - Mg resting on the centre of the bridge. In this case we should have 



Integrating this equation twice, and observing that — = when ^ = c, and »; = when ^ = 0, and 

 that S is the value of t) when ^ = c, we get 



" K--^^ (46) 



Mgh c^ 



Returning now to the bridge in its actual state, we get to determine R', by taking moments about 5, 



B'.2c-/? (2c-,r) + — /" ~ (2c- f) df = (47) 



^c\ df ^ s / v> 



Eliminating B' between (4-5) and (47), putting for K its value given by (46), and eliminating t 



. div 

 by the equation -— = V, we get 



d'ri Sis I, ^,„ M' V- r , „ r^rfS;' , ,„ rf-''^^^', yi.,^r\\ , , 



